Question

The Fourier series coefficient of the signal y(t) = x(t-t0) + x(t+t0) is _____________

(a) 2 cos (\(\frac{2π}{t}\) kt0) X[k]

(b) 2 sin (\(\frac{2π}{t}\) kt0) X[k]

(c) 2 cos (\(\frac{2π}{t}\) kt0)

(d) 2 sin (\(\frac{2π}{t}\) kt0)

I got this question in examination.

This intriguing question comes from Fourier Series Coefficients topic in division Fourier Series of Signals and Systems

Answer 1

Right option is (a) 2 cos (\(\frac{2π}{t}\) kt0) X[k]

For explanation: x (t-t0) is periodic with period T. the Fourier series coefficient of x (t-t0) is X1[k] = \(\frac{1}{T}\) ∫ x (t-t0)e^-jkωt dt

= e^-jkωt0 X[k]

Similarly, the Fourier series coefficient of x (t+t0) is X2[k] = e^jkωt0 X[k]

The Fourier series coefficient of x (t-t0) + x (t+t0) is

Y[k] = X1[k] + X2[k]

= e^-jkωt0 X[k] + e^jkωt0 X[k]

= 2 cos (\(\frac{2π}{t}\) kt0) X[k].